polynomial ring over integral domain

Theorem.

Proof. Let f⁢(X) and g⁢(X) be two non-zero polynomials in R⁢[X] and let af and bg be their leading coefficients, respectively. Thus af≠0, bg≠0, and because R has no zero divisors, af⁢bg≠0. But the productaf⁢bg is the leading coefficient of f⁢(X)⁢g⁢(X) and so f⁢(X)⁢g⁢(X) cannot be the zero polynomial. Consequently, R⁢[X] has no zero divisors, Q.E.D.

Remark. The theorem may by induction be generalized for the polynomial ring R⁢[X1,X2,…,Xn].